## Every ergodic measure is uniquely maximizing

Venue: | Discr. & Cont. Dyn. Sys |

Citations: | 4 - 0 self |

### BibTeX

@ARTICLE{Jenkinson_everyergodic,

author = {Oliver Jenkinson},

title = {Every ergodic measure is uniquely maximizing},

journal = {Discr. & Cont. Dyn. Sys},

year = {},

volume = {16},

pages = {383--392}

}

### OpenURL

### Abstract

Abstract. Let Mφ denote the set of Borel probability measures invariant under a topological action φ on a compact metrizable space X. For a continuous function f: X → R, a measure µ ∈ Mφ is called f-maximizing if � f dµ = sup { � f dm: m ∈ Mφ}. It is shown that if µ is any ergodic measure in Mφ, then there exists a continuous function whose unique maximizing measure is µ. More generally, if E is a non-empty collection of ergodic measures which is weak ∗ closed as a subset of Mφ, then there exists a continuous function whose set of maximizing measures is precisely the closed convex hull of E. If moreover φ has the property that its entropy map is upper semi-continuous, then there exists a continuous function whose set of equilibrium states is precisely the closed convex hull of E. 1. Introduction. Let X be a compact metrizable space, and Γ a topological group or semi-group. Let φ be a topological action of Γ on X, i.e., a continuous map φ: Γ × X → X, (γ, x) ↦ → φγ(x) such that φ1 = idX and φγ ′ ◦ φγ = φγ ′ γ for all γ, γ ′ ∈ Γ. Let B denote the σ-algebra of Borel subsets of X. A Borel probability

### Citations

220 |
Thermodynamic Formalism
- Ruelle
- 1978
(Show Context)
Citation Context ...exists a continuous f such that each element of E is an equilibrium state for f. It follows that every element of the convex hull co(E) = co(E) is also an equilibrium state for f, though the proof in =-=[Rue]-=- does not guarantee that these are the only equilibrium states for f. Acknowledgements. This research was partially supported by an EPSRC Advanced Research Fellowship. I am grateful to Thierry Bousch ... |

168 | An Introduction to Ergodic Theory. Graduate Texts - Walters - 1982 |

101 | and functional analysis, 3rd Edition - Lang, Real - 1993 |

76 | Compact convex sets and boundary integrals - Alfsen - 1971 |

70 | Lectures on Choquet’s theorem - Phelps - 1966 |

24 | Lyapunov minimizing measures for expanding maps of the circle, Ergodic Theory and Dynamical Systems 21 - Thieullen - 2001 |

18 | condition de - Bousch, La |

17 | Le poisson n’a pas d’arêtes, Ann - Bousch |

17 | Frequency locking on the boundary of the barycentre set, Experiment - Jenkinson |

16 | Optimal periodic orbits of chaotic systems occur at low period, Phys - Hunt, Ott - 1996 |

14 | Kazhdan’s property T and the geometry of the collection of invariant measures - Glasner, Weiss - 1997 |

14 | Geometric barycentres of invariant measures for circle maps - Jenkinson |

13 |
Generic properties of invariant measures for Axiom A diffeomorphisms, Invent
- Sigmund
- 1970
(Show Context)
Citation Context ...ic µ being the unique member of some Mmax(f)? Theorem 1 answers this question 3For example, if the action is generated by an Axiom A diffeomorphism T : X → X, then Merg is a proper dense subset of Mφ =-=[Sig]-=-. 4More usually, a closed invariant set is called strictly ergodic if it is both uniquely ergodic and minimal for the action φ. There is an obvious one-to-one correspondence between strictly ergodic m... |

11 | Cohomology classes of dynamically non-negative C k functions, Invent - Bousch, Jenkinson - 2002 |

10 | Locally solid Riesz spaces - Aliprantis, Burkinshaw - 1978 |

10 | Zero temperature limits of Gibbsequilibrium states for countable alphabet subshifts of finite type - Jenkinson, Mauldin, et al. |

8 | The Third Law of Thermodynamics and the Degeneracy of the Ground State for Lattice Systems - Aizenman, Lieb - 1981 |

8 | Silovscher Rand und Dirichletsches Problem - Bauer - 1961 |

7 |
Existence et Unicité des Représentations Intégrales dans les Convexes Compacts Quelconques”, Annales Institut Fourier, Universite de Grenoble (France) 13
- Choquet, Meyer
- 1963
(Show Context)
Citation Context ...be a weak ∗ closed subset of this closed unit ball, so it is itself weak ∗ compact and metrizable. (ii) The fact that Mφ is a simplex is classical, dating back at least as far as the paper of Choquet =-=[Cho]-=-. Since Mφ lies in a hyperplane in E which does not contain the origin, it suffices to show that Eφ = Cφ −Cφ is a sub-lattice 9 of E. This was proved by Choquet [Cho, p. 139–14], but for completeness ... |

7 | Croissance des sommes ergodiques, manuscript, circa 1993. FUNCTIONS ON FLOWERS 23 (a) f −1 (b) g = ϕ − ϕ ◦ T −1 (c) f + g −1 γ γ 2 + γ+ γ 2 + γ+ γ - Conze, Guivarc’h |

7 | Infinite dimensional convexity. In: Handbook on the Geometry of Banach spaces - Fonf, Lindenstrauss, et al. |

7 | Ergodic Theory via Joinings (Mathematical Surveys and Monographs - Glasner - 2003 |

6 |
Séparation des fonctions réelles définies sur un simplexe de
- Edwards
- 1965
(Show Context)
Citation Context ...Define η : Mφ → R by η(ν) = max(λ(ν), 0), ξ : Mφ → R by ξ|F ≡ 0 and ξ| Mφ\F ≡ maxm∈Mφ |λ(m)|. Since η is continuous and convex, ξ is lower semi-continuous and concave, and η ≤ ξ, a theorem of Edwards =-=[Edw1]-=- (cf. [Alf2, Thm. II.3.10]) asserts the existence of a continuous affine functional lµ : Mφ → R such that η ≤ lµ ≤ ξ. In particular, lµ is non-negative because η is, and it vanishes on F since both η ... |

4 |
Some convexity questions arising in statistical mechanics
- Israel, Phelps
- 1984
(Show Context)
Citation Context ...ality. More precisely, our Theorem 1 can be obtained by following the proof of [Phe2, Thm. 1], whose strategy is similar to the one used here, while our Theorem 4 can be obtained from Israel & Phelps =-=[IP]-=- by combining their Propositions 2.1 and 3.9. Reciprocally, some of the results in this paper have analogues in the context of equilibrium states. Notably, the following two theorems can be proved in ... |

3 | Split faces of compact convex sets - Alfsen, Andersen - 1970 |

3 | Unique equilibrium states, Dynamics and Randomness (Santiago, 2000), Nonlinear Phenom - Phelps - 2002 |

3 | Topological Vector Spaces. Graduate Texts in Mathematics - Schaefer - 1971 |

3 | On partially ordered vector spaces and their duals, with applications to simplexes and C∗-algebras - STØRMER - 1968 |

2 | Aliprantis & O. Burkinshaw, Positive operators - D - 1985 |

2 | Effros, Structure in simplexes - G - 1967 |

2 |
La théorie générale de la mesure dans son application à l’étude des systèmes dynamiques de la mécanique non linéaire, Ann
- Krylov, Bogolioubov
- 1937
(Show Context)
Citation Context ...ar, Mφ is non-empty when the action φ is generated by a single map (in which case Γ = Z or Z + ) 2 , or if φ is a flow or semi-flow (in which case Γ = R or R + ), by a theorem of Krylov & Bogolioubov =-=[KB]-=- (see [Wal, Cor. 6.9.1]). Mφ is convex, and when equipped with the weak ∗ topology it is compact and metrizable. A measure µ ∈ Mφ is ergodic if µ(A)(1 − µ(A)) = 0 for every A ∈ B such that µ(A △ φ −1 ... |

2 | Caractérisation des simplexes par des propriétés portant sur les faces fermées et sur les ensembles compacts de points extrémaux - Rogalski - 1971 |

1 | On the geometry of Choquet simplexes - Alfsen - 1964 |

1 | Cônes des fonctions continues sur un espace compact - Boboc, Cornea - 1965 |

1 | A generalized theory of convexity - Davies - 1967 |

1 | Minimum stable wedges of semi-continuous functions - Edwards - 1966 |

1 | Dilated sets and characterizations of simplexes, Invent - Ellis, Roy - 1980 |

1 |
Representation of invariant measures, dittoed notes (17
- Feldman
- 1963
(Show Context)
Citation Context .... 3.3, Cor. 3.5] (see [ER, GdR, Rog, Tay] for related discussion). 9 In fact, Eφ is a Riesz subspace of E. 10 Yet another method of proving that Mφ is a simplex, based on unpublished notes of Feldman =-=[Fel]-=-, can be found in [Phe1, Ch. 10].s388 OLIVER JENKINSON (iii) This is a result of Alfsen [Alf1, Prop. 4] (see also [Alf2, Lem. II.7.18]). (A generalisation of this result (see e.g. [AA, Stø]) is that t... |

1 | de Rugy, Géométrie des simplexes, Centre de documentation universitaire - Goullet - 1968 |

1 | Ergodic optimization, Discrete Continuous Dynam - Jenkinson - 2006 |

1 | The structure space of a Choquet simplex - Taylor - 1970 |